Moderate deviations in a class of stable but nearly unstable processes
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We consider a stable but nearly unstable autoregressive process of any order. The bridge between stability and instability is expressed by a time-varying companion matrix A_n with spectral radius ρ(A_n) < 1 satisfying ρ(A_n) → 1. In that framework, we establish a moderate deviation principle for the empirical covariance only relying on the elements of A_n through 1-ρ(A_n) and, as a by-product, we establish a moderate deviation principle for the OLS estimator when Γ, the renormalized asymptotic variance of the process, is invertible. Finally, when Γ is singular, we also provide a compromise in the form of a moderate deviation principle for a penalized version of the estimator. Our proofs essentially rely on troncations and m--dependent sequences with unbounded m.
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